The theoretical understanding of the IQHE is by now well-developed and there are many excellent reviews and books [1]}.
Some early pioneering works were by Laughlin and Halperin [2]}, [3]}. Let us review some of the very basics.
The pure problem of a single electron in a magnetic field was solved long ago by Landau.
The eigenstates fall into Laudau levels with energy \(E_L(n) = ( n + \tfrac{1}{2}) \hbar e B/m\)  where \(m\)  is the electron mass.
Each Landau level has a large degeneracy of states \(N_{{\rm states}} = \Phi / \Phi _0\)  where \(\Phi \)  is the magnetic flux and \(\Phi _0 = h c/e\) .
These are all de-localized states.
The IQHE is instead a many body problem, namely a finite density electron gas at finite temperature with non-zero \(B\) ,
in the presence of disorder, such as various kinds of random impurities or defects. The plateaux would not exist without this disorder
and a complete understanding involves Anderson localization. Since we will not be considering any detailed physically motivated
hamiltonian,
let us just give a rough and brief picture of the phenomenon. At energies approximately in the gap between the pure Landau levels the states are localized due to disorder and thus don't contribute to the conductivity and it remains constant on the plateaux. At certain critical energies \(E^c_n\)  near the original Landau levels, where by definition
\(E^c_{n+1} > E^c_n\) , there is a quantum phase transition where some states are delocalized and contribute to a change of the conductivity.
Let us write this in terms of the Fermi energy \(E_F\) :
\({\rm If} ~~ E^c_{n} > E_F > E^c_{n-1} ~~~~{\rm then } ~~ \rho _{xy} = \dfrac{1}{n} \, \dfrac{h}{e^2}\) 
